Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method, Draft
Wilcox, Lucas C.
MetadataShow full item record
This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate computation of these derivatives is important, for instance, in inverse problems and optimal control problems. This computation is usually based on an adjoint PDE system, and the question addressed in this paper is how the discretization of this adjoint system should relate to the dG discretization of the hyperbolic state equation. Adjoint-based derivatives can either be computed before or after discretization; these two options are often referred to as the optimize-then-discretize and discretize-then-optimize approaches. We discuss the relation between these two options for dG discretizations in space and Runge– Kutta time integration. The influence of different dG formulations and of numerical quadrature is discussed. Discretely exact discretizations for several hyperbolic optimization problems are derived, including the advection equation, Maxwell’s equations and the coupled elastic-acoustic wave equation. We find that the discrete adjoint equation inherits a natural dG discretization from the discretization of the state equation and that the expressions for the discretely exact gradient often have to take into account contributions from element faces. For the coupled elastic-acoustic wave equation, the correctness and accuracy of our derivative expressions are illustrated by comparisons with finite difference gradients. The results show that a straightforward discretization of the continuous gradient differs from the discretely exact gradient, and thus is not consistent with the discretized objective. This inconsistency may cause difficulties in the convergence of gradient based algorithms for solving optimization problems.
Showing items related by title, author, creator and subject.
Finite volume element (FVE) discretization and multilevel solution of the axisymmetric heat equation Litaker, Eric T. (Monterey, California. Naval Postgraduate School, 1994-12);The axisymmetric heat equation, resulting from a point-source of heat applied to a metal block, is solved numerically; both iterative and multilevel solutions are computed in order to compare the two processes. The continuum ...
Renick, Dirk H. (2001-06);One of the main problems affecting modern propulsor design engineers is the ability to quantitatively predict unsteady propeller forces for modern, multi-blade row, ducted propulsors operating in highly contracting flowfields. ...
Coupled Azimuthal Potentials for Electromagnetic Field Problems in Inhomogeneous Axially Symmetric Media Morgan, Michael A.; Chang, Shu-Kong; Mei, Kenneth K. (1977-05);Classical electromagnetic potential formulations are, with the exceptions of a few special cases of one-dimensional stratification, restricted to use in uniform media. A recently developed potential formulation that ...